Descriptive Set Theory and Computability

An important problem encountered throughout mathematics is to completely classify some type of mathematical object by invariants. The field of descriptive set theory gives a general framework for studying these types of classification problems and comparing their relative difficulties. The PI proposes research in descriptive set theory and its connections with other mathematical fields including computability, operator algebras, topological dynamics, and ergodic theory. The PI will continue facilitating connections with these mathematical communities, and engaging with graduate students and young researchers. The project will support the training of graduate students at UCLA. The PI proposes research on Weiss's question on amenability and hyperfiniteness using tools from Gromov's theory of asymptotic dimension. This approach to Weiss's question has already greatly extended and simplified prior results on the problem and clarified their proofs. This investigation has applications to topological dynamics and operator algebras. The PI also proposes research on classical geometrical paradoxes such as the Banach-Tarski paradox and Tarski's circle squaring problem. Recent advances in measurable combinatorics have led to new theorems giving measurable solutions to these problems using combinatorial techniques from the study of flows and matching problems on Borel graphs. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.